Math 142, Chapter 5
Spring 2011
Zarestky
Chapter 5 Reading Notes
In this chapter, we will learn about applications of the first and second derivative to graphs. Then, we will
apply our knowledge of derivatives to solve optimization (maximization and minimization) problems.
Section 5.1: First Derivatives and Graphs
By the end of this section, you should be able to:
• Analyze the graph of a function.
o Identify the intervals on which a function is increasing or decreasing.
o Determine the intervals on which the first derivative is positive or negative.
• Determine local extrema and intervals of increasing and decreasing using the first derivative.
o Identify the critical values of a function.
o Create a sign chart of the characteristics of the first derivative.
• Match the graph of a function to the graph of its derivative.
In the chapter 3, we learned about the first derivative as the slope of a graph. In this section, we will
explore the relationship between the first derivative and the shape of the graph in general.
Here is the basic idea:
1. Calculate the first derivative.
2. Set the derivative equal to 0 and solve for x. (I.e., locations of horizontal tangent lines.)
3. Identify any values of x for which the derivative doesn’t exist.
4. Use all the x’s to make a sign chart and note where the derivative is positive or negative.
5. Intervals on which the first derivative is positive or negative correspond to intervals on which the
original function is increasing or decreasing.
Work Example 1. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
Make a note of the definition of critical values. A critical number is always in the domain of the original
function.
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Work Example 3. Pay particular attention to the difference between continuity and differentiability.
Compare your answers to the book’s solution. What questions do you have?
Work Example 4. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
Work Example 5. Compare your answers to the book’s solution. What questions do you have?
Read the paragraphs about Local Extrema. Make some notes in your own words about the definitions of
the new vocabulary words: local maximum, local minimum, local extremum, and turning point.
Work Example 6. Compare your answers to the book’s solution. What questions do you have?
The first derivative test builds off of the sign charts you already know how to make. Spend a few
moments looking over the 4 cases in the blue box and the graphs of examples on p. 274.
Work Example 7. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
Work Example 8. Compare your answers to the book’s solution. What questions do you have?
Work Example 9. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-18, 19-39(odd), 47-61(odd), 63-68, 69-93(odd).
Do you have any questions?
Math 142, Chapter 5
Spring 2011
Zarestky
Section 5.2: Second Derivative and Graphs
By the end of this section, you should be able to:
• Analyze the graph of a function.
o Identify the intervals on which a function is concave up or concave down.
o Determine the intervals on which the second derivative is positive or negative.
• Calculate higher-order derivatives.
• Determine the point of diminishing returns of a total sales model.
• Determine the inflection points and concavity of a function.
o Create a sign chart of the characteristics of the second derivative.
In this section, we will build off of the relationship between the first derivative and the shape of the graph
of the original function to explore the relationship between the second derivative and the shape of the
graph. The first derivative gave information about increasing and decreasing behavior. The second
derivative describes concavity.
Skip the definition on p. 285 and focus on the summary on p. 286. Spend a few moments looking over the
examples of graphs on p. 286.
Work Example 1. Compare your answers to the book’s solution. What questions do you have?
The next part of this section describes inflection points. You can use inflection points and second
derivatives in a very similar manner to critical values and first derivatives.
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
Work Example 3. Compare your answers to the book’s solution. What questions do you have?
Now that you have both first and second derivatives in your mathematical tool belt, combine them to
analyze graphs and sketch curves without a calculator. Scary, I know. 
Work Example 4. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
Work Example 5. Compare your answers to the book’s solution. What questions do you have?
Work Example 6. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
One application of the second derivative is the point of diminishing returns. Make a note of the new
vocabulary.
Work Example 7. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-6, 7-67(odd), 77, 79, 89, 91.
What questions do you still have?
Math 142, Chapter 5
Spring 2011
Zarestky
Section 5.4: Curve-Sketching Techniques
By the end of this section, you should be able to:
• Sketch the graph of a function given information about the first and second derivatives.
• Sketch the graph of a rational function.
o Determine the domain, intercepts, and asymptotes of the function.
o Analyze and unite the properties of the first and second derivatives by using sign charts.
In this section, you will build on the curve sketching skills you have already started to develop. Use the
given graphing strategy for the examples in this section. Curve sketching can be challenging. It is
essential that you get a lot of practice!
Work Examples 1 through 5 on your own paper. Compare your answers to the book’s solution for each.
What questions do you have? (In example 5, don’t stress about oblique asymptotes.)
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-9(odd), 17-55(odd), 65-71(odd), 77, 79.
What questions do you still have?
Math 142, Chapter 5
Spring 2011
Zarestky
Section 5.5: Absolute Maxima and Minima
By the end of this section, you should be able to:
• Determine the absolute maximum and minimum values of a function on a given interval.
You have already used derivatives to analyze local extrema. In this section, you will continue the process
to analyze absolute extrema. How are absolute extrema different than local extrema? Make some notes
about the extreme value theorem.
Work Example 1. Compare your answers to the book’s solution. What questions do you have?
Apply the second derivative test from p. 327 and 329 in the next examples.
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 5
Spring 2011
Zarestky
Work Example 3. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-69(odd).
What questions do you still have?
Math 142, Chapter 5
Spring 2011
Zarestky
Section 5.6: Optimization
By the end of this section, you should be able to:
• Solve real-world optimization problems.
Optimization problems are problems in which you are asked to find an absolute maximum or minimum
value. These problems can be long. Take your time and use your logic and reasoning skills.
Work Examples 1 through 8 on your own paper. Compare your answers to the book’s solutions. Make
notes about any questions you have.
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-13(odd), 17-25(odd).
What questions do you still have?